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and
EQUILIBRIUM OF A PARTICLE                     25
= F_ __ P cos a — W sin a ^ N     P sin a. + W cos a
p
DISCUSSION. — (a)   When   a = 0,   M = —,   and   R = W *
W
(b) When P = 0, ju = —tan a; therefore a = — <j>, that is, the inclined plane must be tipped in the opposite direction and must be given an angle of elevation equal to the angle of friction in order that motion may take place towards the positive direction of the x-axis.
PROBLEMS.
1.   A body which weighs 100 pounds is barely started to move on a-rough horizontal plane by a force of 150 pounds acting in a direction making 30° with the horizon.   Find R and At.
2.   A body placed on a rough inclined plane barely starts to move.' when acted upon by a force equal to the weight of the body.   Find the coefficient of friction, (a) when the force is normal to the plane; (b) when it is parallel to the plane.
3.   A horizontal force equal to the weight of the body has to be applied in order to just start a body into motion on a horizontal floor.   Find the coefficient of friction.
4.   A weight W rests on a rough inclined plane, which makes an angle a. with the horizon.   Find the smallest force which will move the weight if the coefficient of friction is AC.
6. How would you determine experimentally the coefficient of friction between two bodies?
6.   A weight of 75 pounds rests on a rough horizontal floor.   Find the magnitude of the least horizontal force which will move the body if the coefficient of friction is 0.4; also find the reaction of the plane.
7.   A particle of weight W is in equilibrium on an inclined plane under the action of a force F, which makes the magnitude of the normal pressure equal W.   The coefficient of friction is At and the angle of elevation of the inclined plant) is a.   Find the magnitude and direction of the force.
8.   An insect starts from the highest point of a sphere and crawls down.   Where will it begin to slide if the coefficient of friction between; the insect and the sphere is |?                                                            """
9.   The greatest force, which can keep a particle at rest, acting alongr an inclined plane, equals twice the least force.   Find the coefficient or: friction.   The angle of elevation of the plane is a.                           ^ -3
31.  Resultant of a System of Forces. — The resultant of & number of forces which act upon a particle is a force wlfieS